Rotary piston engine with improved housing and piston configuration

ABSTRACT

A rotary piston engine has a plurality of explosion chambers situated around the periphery of the housing and includes within the housing a piston. According to the preferred embodiment, a housing having H recesses would contain a piston having P = H-1 lobes. According to the preferred embodiment, the structure of the housing relative to the piston, and vice-versa, has been improved to give optimum results. The relationship between the housing and the piston is such that it may be characterized by an optimum mathematical relationship which can make the construction of such machinery simpler than has been heretofore known. Additionally, the relationship between the improved piston and housing configuration is related to other engine parameters such as the compression ratio and efficiency.

RELATED APPLICATIONS

This invention represents an improvement over the invention described inapplication Ser. No. 194,196, filed Nov. 1, 1971 and entitled "ROTARYPISTON WITH MULTI-EXPLOSION CHAMBERS", now U.S. Pat. No. 3,771,501issued Nov. 13, 1973.

BACKGROUND OF THE INVENTION

1. Field of the Invention

This invention relates to internal combustion engines in general, and inparticular to a rotary piston engine having an improved housing andpiston configuration.

2. Description of the Prior Art

This invention discloses an improvement in the invention discussed in myU.S. Pat. No. 3,771,501 issued Nov. 13, 1973. While rotary pistonengines of the sort disclosed in my previous patent are advantageous inmany ways over other prior art combustion engines, it has heretoforebeen difficult to construct and fabricate these engines without a lot oftrial and error. In order to improve the ease of construction of suchdevices a new relationship is proposed between the piston and the outerhousing which will allow for simpler manufacture of parts. Additionally,the relationship between piston and its eccentric drive has beendiscovered which allows an eccentric drive mechanism to be constructedeasily and inexpensively. Therefore, many problems posed by the priorart can be overcome by the use of the hereinafter described invention.

SUMMARY OF THE INVENTION

Briefly, in the preferred embodiments disclosed herein, a rotary pistonengine is described in which the relationship between the piston and thehousing is such that maximum advantages can be obtained from thisconstruction. In particular, both the piston and the housing can bedescribed in terms of an epitrochoid. If the housing is constructedaccording to an epitrochoid curve, then a piston can be constructedwhich will most readily fit and function within that housing.Conversely, for a given epitrochoid piston, a most suitable epitrochoidhousing may be fabricated. It has been found that the relationshipbetween the piston and the housing can be described in terms of fiveindependant equations in eight unknowns. Given this situation it ispossible to construct the best piston or best housing for a given set ofparameters. Obviously, if three of the eight unknowns can be specifiedthen the other five unknowns can be solved by means of the fivesimultaneous equations. Frequently, considerations such as the desirednumber of lobes, the ultimate size of the housing and the desired horsepower will dictate several of the unknowns thereby leaving thefabricator with an easy means of solving for the other unknownquantities.

It can be shown mathematically that different compression ratios can beobtained by selecting appropriate values for the radius of the interiorof the housing (R) and the eccentricity (e) of the drive mechanism. Itis further possible to determine the maximum ratio of R to e to providethe maximum compression or horse power as desired. In a manner similarto the foregoing, an optimum relationship is proposed between theeccentric drive gearing system and the piston which will allow forimproved construction efficiency and simplicity.

These and other advantages of the present invention will be more fullyunderstood in view of the accompanying drawings.

DESCRIPTION OF THE DRAWINGS

FIG. 1 is a cross section view of the preferred embodiment of thepresent invention;

FIG. 2 is a detailed cross sectional view of the multi-recessesstationary housing;

FIG. 3 is a view of the outline of a multi-lobe rotary piston which isreceived within the stationary housing of FIG. 1;

FIG. 4 is a cross sectional view of the invention according to apreferred embodiment in which the outline of the configuration of theeccentric gearing system is illustrated;

FIG. 5 is a vector diagram of the locus of the point p_(p) (x_(p),y_(p)); and

FIG. 6 is a simplified version of the diagram in FIG. 5.

DESCRIPTION OF THE PREFERRED EMBODIMENT

With reference to the following description it will be appreciated thatlike numbers and like letters will refer to similar elements as shown inthe different views of FIGS. 1 - 4.

A profile of the piston and the housing assembly according to apreferred embodiment of the present invention can be found withreference to FIG. 1. The recesses of the housing H and the lobes of thepiston P can be expressed in terms of mathematical equations whichgenerate epitrochoid curves. An important facet of this invention is thespecific relationship between the shape of the piston and the shape ofthe housing given such an epitrochoid configuration. Before dealing withthe specifics of the piston to housing relationship, the followingbackground of history dealing with rotary piston engines should beunderstood.

According to the drawings, there is illustrated a rotary piston enginewith dual explosion chambers, in which the stationary housing has Hrecesses and the rotary piston has P = H -1 lobes. It will be noted withreference to FIG. 1 that the number of recesses of the housing H = 4 andthat therefore the number of lobes of the piston P = 4 - 1 = 3. Thecurves generated in FIG. 1 are compound curves of high degree and aredesignated as C and C₁. It will be appreciated with respect to thedescribed rotary piston engine that more than dual explosion chamberscan be constructed using the same principles (not shown).

In general, the efficiency of a rotary piston engine with multipleexplosion chambers depends upon the compression ratio E which is arelationship between the maximum volume V_(max). and the minimum volumeV_(min).. V_(max). is the maximum volume trapped between theeccentrically rotating piston surface and the stationary housingsurface. The seals on the piston determine the size of the sealedcavity. Accordingly, V_(min). is the minimum volume trapped between thepiston lobes and the engine housing. From this relationship it can beshown that: ##EQU1## where E = Compression ratio.

V_(max). = Maximum volume.

V_(min). = Minimum volume.

According to the rotary piston engine of the present invention, thecompression ratio E is limited by the rotary piston curve profilegenerating radius R and the eccentricity e of the rotary piston. Morespecifically, the compression ratio E is restricted by the ratio of R toe. This relationship is expressed in the following equation:

    K = R/e                                                    (2)

where

K = Ratio factor.

R = Rotary piston curve profile generating radius.

e = Eccentricity of the rotary piston.

With respect to the housing it can also be shown that the stationaryhousing curve profile generating radius R₁ = R + e and that theeccentricity e of the rotary piston can be combined in the followingrelationship: ##EQU2## where K₁ = Ratio Factor.

R₁ = R + e = Stationary housing curve profile generating radius.

e = Eccentricity of the rotary piston.

It is clear from the foregoing that given the rotary piston curveprofile generating radius R and the eccentricity of the rotary piston eit is possible to define the stationary housing curve profile generatingradius R₁ from the relationship R₁ = R + e. Additionally, in view of theforegoing it can be shown that:

    K.sub.1 = K + 1                                            (4)

the special significance of the foregoing relationships will be dealtwith in more detail hereinafter.

A detailed description of the profile of the housing cavity is shown inFIG. 2. Essentially the profile C is an epitrochoid which is generatedas a small circle with diameter D₁ rolls around the periphery of alarger circle with a diameter of D_(o). The profile C is the locus ofthe point p_(h) (x_(h),y_(h)) as the circle with diameter D₁ revolvesaround the circle with diameter D_(o). In other words, the profile C isthe locus of points generated by point p_(h) (x_(h),y_(h)) as the pointp_(h) (x_(h),y_(h)) moves through 360° of arc.

It will be evident from observing the contous of the housing interior ofFIG. 2 that in order for the curve to be smooth and continuous, it isnecessary that point p_(h) (x_(h),y_(h)) return to its originating pointafter 360° of revolution. In order for point p_(h) to return to itsoriginal starting place, it is necessary that the circle with diameterD₁ revolve at integral number of times. In the example of FIG. 2 where H= 4 it is obvious from inspection that the circle with diameter D₁revolves exactly five times with respect to the coordinate axis of thecircle with diameter D_(o). However, with respect to the periphery ofthe circle with diameter D_(o), the circle with diameter D₁, of course,only revolves four times. In other words, the circle with diameter D₁revolves exactly once for every 72° from arc. Stated another way, it isclear that in order to generate a smooth, continuous curve thecircumference of the circle with diameter D₁ must be devisable into thecircumference of the diameter of the circle with diameter D_(o) with aresultant that is a whole integral number. That whole, integral is equalto the number of recesses H of the housing, and the followingrelationship is readily apparent from inspection:

    H = D.sub.o  /D.sub.1                                      (5)

another convenient way to describe the contour C of the housing profileis in terms of a minimum radius R and an eccentricity factor e. Theminimum interior radius R is defined as the minimum distance between thecenter of the curve C and the closest portion of the curve C to thecenter 0. R may also be defined and will be shown later to be equal tothe piston curve profile generating radius. The eccentricity factor erepresents the displacement of the piston during its cyclical travel.The following relationship will also be clear from inspection of FIG. 2:##EQU3##

In the context of this invention the circle with diameter D₁ is referredto as the rolling circle and the circle with diameter D_(o) is referredto as the fixed, or stationary, circle. The curve C is generated as therolling circle revolves around the fixed circle.

When the chosen point p_(h) (x_(h),y_(h)) is on the circumference of therolling circle with D₁ diameter, the curve generated thereby is calledan epicycloid. The interior points generated by an epicycloid tend to berather sharp and may not be desirable in a rotary piston engine. As theeccentric distance e is decreased relative to the diameter D₁ of therolling circle, the points on curve C closest to the origin O becomesmoother. The curve generated when the point p_(h) (x_(h),y_(h)) iswithin the diameter D₁ of the rolling circle but not on the periphery ofthe circle, is called an epitrochoid. Stated another way:

    If D.sub.o /D.sub.1 = R/e = K                              (7)

then the curve is an epicycloid.

Conversely, if

    D.sub.o /D.sub.1 < (R/e = K)                               (8)

then the curve is an epitrochoid.

Where

D_(o) = A fixed, stationary circle, for stationary housing.

D₁ = Rolling circle diameter, for stationary housing.

R = Piston curve profile generating radius or minimum radius of housingcurve C.

e = Eccentricity of the rotary piston; and

K = R/e = Ratio factor.

In general, where H is less than or equal to 8, it is desirable toconstruct the stationary housing curve profile in the form of anepitrochoid. In the event where H is equal or greater than 9, it may beeconomically and technically desirable to make the profile of thestationary housing curve in the form of an epicycloid.

With reference to FIG. 3 the profile of the piston element may begenerated in a similar fashion as the profile for the housing shown inFIG. 2. The profile of the piston is a curve C₁ generated by the locusof a point p_(p) (x_(p),y_(p)) located within the diameter D₃ of arolling circle as it revolves through 360° of arc around a fixed circleof diameter D₂. In this manner, the generation of the profile curve ofthe piston is directly analogous to the method of generating the profilecurve of the housing. In practice it has been found that the optimumpiston size is achieved where the number of lobes P of the piston is oneless than the number of recesses H of the housing.

It should be evident that while H = P + 1 is the most advantageousrelationship, and it is practical to construct pistons in which H isexactly one whole integer larger than P. It will be appreciated that Hand P, however, are always whole integral numbers.

As with the housing profile of FIG. 2, the piston profile C₁ may also bedescribed in terms of the generating radius R and the eccentricity e. Itwill be clear from inspection and a careful review of the foregoingthat: ##EQU4##

It is also clear, and especially in view of the discussion with respectto the housing profile, that in order to generate a smooth, continuouscurve C₁ it is necessary that:

    P = D.sub.2 /D.sub.3                                       (10)

another way to express some of the foregoing relationships is by theequation: ##EQU5## however it will be appreciated that the foregoingrelationship is not an independant relationship, but is instead a ratiobetween two previously discussed relationships.

A review of the relationship between housing profile C and pistonprofile C₁ will show that the optimum piston curvature for a givenhousing, or vice versa, can be expressed in terms of the following fiveindependant equations:

    I. H = P + 1;

    ii. h = d.sub.0 /d.sub.1 ;

    iii. p = d.sub.2 /d.sub.3 ; ##EQU6## Where the eight parameters are: H = The number of recesses on the stationary housing (the same as the number of convex corners);

P = The number of lobes on the piston (the same as the number of convexcorners);

R = Minimum interior radius of the housing (the same as the piston curveprofile generating radius);

e = The eccentricity of the rotary piston;

D₀ = Fixed circle diameter of the stationary housing;

D₁ = Rolling circle diameter of the stationary housing;

D₂ = Fixed circle diameter of the rotary piston; and

D₃ = Rolling circle diameter of the rotary piston.

In essence, the relationship between the piston profile and the housingprofile can be described in terms of five independant equations in eightunknowns. It is deductable from elementary simultaneous equationmathematics, that if any three of the eight unknowns are known or given,then the other five perimeters can be derived from the five independantrelationships. For example:

Given that:

R = 10 inches;

e = 2 inches; and

P = 3

it is then possible to solve for the remaining values of H, D₀, D₁, D₂and D₃ as follows:

STATIONARY HOUSING PARAMETERS

    1. H = P + 1

    1a. H = 3 + 1 = 4 ##EQU7##

    4. H = D.sub.o /D.sub.1 ##EQU8## Where H = The number of recesses on the stationary housing.

D₀ = Fixed circle diameter of the stationary housing.

D₁ = Rolling circle diameter of the stationary housing.

R + e = Generating radius of the stationary housing curve profile.

ROTARY PISTON PARAMETERS

    1. P = H - 1

    1a. P = 4 - 1 = 3 ##EQU9##

    4. P = D.sub.2 /D.sub.3

    4a. P = 15/5 = 3 ##EQU10## Where P = The number of lobes on the rotary piston.

D₂ = Fixed circle diameter of the rotary piston.

D₃ = Rolling circle diameter of the rotary piston.

R = Generating radius of the rotary piston curve profile.

Q.e.d.

in actual practice, it has been found that many of the parameters may begiven by the circumstances surrounding the use of the engine. Forinstance, if low horse power or small sizes necessary, then the ultimatediameter of the housing will be a known factor. Additionally, for avariety of reasons, it may be desirable to build a piston with a minimumnumber of lobes. For instance, in order to cut down on unnecessaryignition circuits and manufacturing costs, it may be desirable to buildthe engine with only four housing recesses as illustrated in FIGS. 1 -4. Another factor, that may be given, is the desired compression factorwhich in turn is related to an optimum R/ e ratio. Therefore, for agiven engine size and horsepower and for a desired compression ratio, itis possible to construct a rotary piston engine having the optimalpiston to housing profile.

It is evident that in creating the stationary housing curve profile, thegenerating radius R₁ = R + e and the required eccentricity e of therotary piston are primary factors. The rolling circle with D₁ diameteras it constantly follows the circle resulted from the generating radiusR₁ = R + e dictates the proper positions of the generating point P_(h)(x_(h),y_(h)). In other words if the generating radius R₁ is designateda hypotenuse with an angle h relative to the origin O of the cartesiancoordinate system, then the eccentricity e of the rotary piston as asecond generating radius becomes a hypotenuse with an angle α_(h1)relative to the center of the rolling circle with D₁ diameter anddetermines the proper positions of the generating point P_(h)(x_(h),y_(h)) on the stationary housing curve profile C in a cartesiancoordinate system. Each position of the generating point p_(h)(x_(h),y_(h)) has a suitable angle α_(h) and α _(h1). The followingequation discloses how α_(h) and α_(h1) are computed: ##EQU11##

    α.sub.h1 = (H + 1) . α.sub.h                   (13)

Where

α_(h) = generating angle with the hypotenuse of the generating radius R₁= R + e

α_(h1) = generating angle with the hypotenuse of the eccentricity e ofthe rotary piston.

The following equations disclose the preferred relationship between thefixed, stationary circle with D₀ diameter and the rolling circle with D₁diameter in connection with related parameters: ##EQU12##

    D.sub.o = H . D.sub.1                                      (15)

    D.sub.1 = D.sub.o /H                                       (16) ##EQU13##

    r = r.sub.1 - e                                            (18)

    e = R.sub.1 - R                                            (19)

    h = d.sub.o /D.sub.1                                       (20)

where

D_(o) = Diameter of the fixed, stationary circle, for stationaryhousing;

D₁ = Diameter of the rolling circle, for stationary housing;

R₁ = R + e = Stationary housing curve profile generating radius;

R = Rotary piston curve profile generating radius;

e = Eccentricity of the rotary piston;

H = Number of the geometrical convex recesses of the stationary housing.

The following equations disclose the relationship of x_(h) and y_(h) tothe cartesian coordinate system:

    x.sub.h = cos α.sub.h . R.sub.1 + cos α.sub.h1 . e (21)

    x.sub.h = cos α.sub.h . (R + e) + cos (H + 1)α.sub.h . e (22)

    y.sub.h = sin α.sub.h . R.sub.1 + sin α.sub.h1 . e (23)

    y.sub.h = sin α.sub.h . (R + e) + sin (H + 1)α.sub.h . e (24)

    α.sub.h = 0° to 360°                   (25)

    α.sub.h1 = (H + 1) . α.sub.h = 0° to (H . 360° + 360°)                                              (26)

where

y_(h) = Abscissa, function;

y_(h) = Ordinate, function;

α_(h) = Generating angle with the hypotenuse of the generating radius R₁= R + e . ;

α_(h1) = Generating angle with the hypotenuse of the eccentricity e ofthe rotary piston;

R = Rotary piston curve profile generating radius;

R₁ = Stationary housing curve profile generating radius;

e = Eccentricity of the rotary piston.

The following explanation and relationships will show how to develop thearea A_(h) of the stationary housing. From the preceding explanation andrelationships, it is evident that a point p_(h) (x_(h),y_(h)) on thestationary housing curve profile is a function of x_(h) and y_(h). Theforegoing relationships disclose the positions of the point p_(h)(x_(h),y_(h)) which can be described in terms of a triangle withabscissa x_(h) and ordinate y_(h) or a hypotenuse of polar radius y_(h)with an angle θ_(h) relative to the origin 0. The equations of thetriangle are the following:

    sin θ.sub.h = y.sub.h /r.sub.h                       (27)

    cos θ.sub.h = x.sub.h /r.sub.h                       (28)

    r.sub.h = y.sub.h /sin θ.sub.h                       (29)

    r.sub.h = x.sub.h /cos θ.sub.h                       (30)

    r.sub.h = √x.sub.h.sup.2 + y.sub.h.sup.2            (31)

    r.sub.h = [R.sub.1.sup.2 + e.sup.2 + 2 Re cos HaH].sup.1/2 (32)

where

    θ.sub.h = Polar angle.

    r.sub.h = Hypotenuse or polar radius.

    x.sub.h = Abscissa, function.

    y.sub.h = Ordinate, function.

The following relationships disclose the area A_(h) of the stationaryhousing of a rotary piston engine with multi-explosion chambers:##EQU14##

The following relationships disclose the area A_(hs) of the circularsection of the stationary housing: ##EQU15## Where θ_(h) = Polar angle.

θ_(1h) = Circular sectors angle, lower limit.

θ_(2h) = Circular sectors angle, upper limit.

The equations below disclose the lengths of the curve profile S_(h) ofthe stationary housing: ##EQU16##

The following equations disclose the lengths of the curve profile S_(hs)of the circular section of the stationary housing: ##EQU17##

The transverse cross-sectional detail view of the multi-shaped rotarypiston profile (FIG. 3) illustrates an algebraic curve of high degree.

According to FIG. 3, an algebraic curve of high degree will beconstructed by using a fixed, stationary circle with D₂ diameter and arolling circle with D₃ diameter. In the rolling circle with D₃ diameter,a point P_(p) (x_(p), y_(p)) will be chosen if the rolling circle withD₃ diameter is rolled around the outside of the stationary circle withD₂ diameter without sliding, then the point p_(p) (x_(p), y_(p)) createsan algebraic curve of high degree.

When the choisen point p_(p) (x_(p), y_(p)) is on the circumference ofthe rolling circle with D₃ diameter, then the curve C₁ generated by therolling circle with D₃ diameter as it rolls on the exterior of the fixedcircle with D₂ diameter is called an epicycloid.

If the generating point p_(p) (x_(p), y_(p)) is within the circumferenceof the rolling circle with D₃ diameter, then the curve C₁ generated iscalled an epitrochoid.

Therefore, if: ##EQU18## then the curve C₁ is called an epicycloid.

If: ##EQU19## then the curve C₁ is called an epitrochoid. Where:

D₂ = Fixed, stationary circle, for rotary piston curve profile;

D₃ = Rolling circle, for rotary piston curve profile;

R = Rotary piston curve profile generating radius;

e = Eccentricity of the rotary piston;

K₂ = Ratio factor.

In general terms where p ≦ 7 the rotary piston curve profile should beconstructed using the method for an epitrochoid. If p ≧ 8 then therotary piston curve profile could be constructed using the method for anepicycloid, taking into consideration other technical and economicfacts.

The following summarizes facts about the K, K₁ and K₂ ratios relative toother parameters:

K = R/e (55 ) ##EQU20## Where: K = Ratio factor, middle;

K₁ = Ratio factor, upper;

K₂ = Ratio factor, lower;

R = Rotary piston curve profile generating radius;

R₁ = Stationary housing curve profile generating radius;

R₂ = Differential radius;

e = Eccentricity of the rotary piston.

It is therefore clear that K₁ > K > K₂.

The ratio factors K, K₁ and K₂ are used to refer to a graduation ofratios which mathmatically dictate restrictions on the compression ratioE. The ratio factors are also determined by the curve profiles of thestationary housing and the rotary pistons as well as by the practicallimits inherent in internal combustion engines.

Using the above facts, the following relationships can be computed:##EQU21##

    K.sub.1 = K + 1 = K.sub.2 + 2                              (59 )

    k.sub.2 = k - 1 = k.sub.1 - 2                              (60 )

it is evident, that in creating the rotary piston curve profile, thegenerating radius R and the required eccentricity e of the rotary pistonare primary factors. As the rolling circle with D₃ diameter follows thecircle resulting from the generating radius R, the rotary piston curveprofile is dictated by generating point p_(p) (x_(p), y_(p)) with theeccentricity e of the rotary piston as a second generating radius. Inother words, if the generating radius R is designated a hypotenuse withan angle α_(p) relative to the origin O₁ of the cartesian coordinatesystem, and the eccentricity e of the rotary piston as a secondgenerating radius is designated a hypotenuse with an angle α_(p1), acurve C₁ may be generated by the locus of points p_(p) (x_(p), y_(p)).Each position of the generating point p_(p) (x_(p), y_(p)) has acorresponding angle α_(p) and an angle α_(p1). The following equationsdisclose how it is computed: ##EQU22##

    α.sub.p1 = (P + 1 ) . α.sub.p                  (62 )

Where:

α_(p) = Generating angle with the hypotenuse of the generating radius R;

α_(p1) = Generating angle with the hypotenuse of the eccentricity e ofthe rotary piston.

The following equations are developed for describing the fixed,stationary circle with D₂ diameter and the rolling circle with D₃diameter with respect to related factors: ##EQU23##

    D.sub.2 = P . D.sub.3                                      (64 ) ##EQU24##

    D.sub.3 = D.sub.2 /P                                       (66 ) ##EQU25##

    R.sub.1 = R + e                                            (68 )

    R.sub.2 = R - e = R.sub.1 - 2 . e                          (69 ) ##EQU26##

    P = D.sub.2 /D.sub.3                                       (71 )

Where:

D₂ = Fixed, stationary circle, for rotary piston;

D₃ = Rolling circle, for rotary piston;

R = Rotary piston curve profile generating radius;

R₁ = Stationary housing curve profile generating radius;

R₂ = Differential radius;

e = Eccentricity of the rotary piston;

P = Number of the geometrical convex corners of the rotary piston.

The equations below describe functions of x_(p) and y_(p) :

    x.sub.p = cos α.sub.p . R + cos α.sub.p1 . e   (72 )

    x.sub.p = cos α.sub.p . R + cos (P + 1 ) α.sub.p . e (73 )

    y.sub.p = sin α.sub.p . R + sin α.sub.p1 . e   (74 )

    y.sub.p = sin α.sub.p . R + sin (P + 1 ) α.sub.p . e (75 )

    α.sub.p = 0° to 360°                   (76 )

    α.sub.p1 = (P + 1 ). α.sub.p = 0° to (P . 360° + 360° )                                             (77 )

where:

x_(p) = Abscissa, function;

y_(p) = Ordinate, function;

α_(p) = Generating angle with the hypotenuse of the generating radius R;

α_(p1) = Generating angle with the hypotenuse of the eccentricity e ofthe rotary piston;

R = Rotary piston curve profile generating radius;

e = Eccentricity of the rotary piston.

The following explanation and relationships disclose the area A_(p) ofthe rotary piston in a rotary piston engine with multi-explosionchambers. From the preceding explanation and the correspondingrelationships it is evident that the point p_(p) (x_(p), y_(p)) on therotary piston curve profile is a function of x_(p) and y_(p). Theforegoing equations describe the positions of the point p_(p) (x_(p),y_(p)) which form a triangle with abscissa x_(p) and which may bedescribed as an ordinate y_(p) hypotenuse or the polar radius r_(p) withan angle θ_(p) relative to the origin O₁. The relationships of thistriangle are as follows:

    sin θ.sub.p = y.sub.p /r.sub.p                       (78 )

    cos θ.sub.p = x.sub.p /r.sub.p                       (79 )

    r.sub.p = y.sub.p /sin θ.sub.p                       (80 )

    r.sub.p = x.sub.p /cos θ.sub.p                       (81 )

    r.sub.p = √x.sub.p.sup.2 + y.sub.p.sup.2            (82 )

    r.sub.p = [R.sup.2 +e.sup.2 + 2.R.e.cos (pα.sub.p)].sup.1/2(83 )

where:

    θ.sub.p = Polar angle.

    r.sub.p = Hypotenuse or polar radius.

    x.sub.p = Abscissa, function.

    y.sub.p = Ordinate, function.

FIG. 5 is a diagram of the vectors associated with the point p_(p)(x_(p), y_(p)). The point p_(p) (x_(p), y_(p)) is located at the tip ofpolar radius vector r_(p). Vector r₁ is the resultant vector of thegenerating radius vector R and the piston excentricity vector e. FIG. 6is a simplified version of FIG. 5. From the foregoing relationships itis relatively easy to define other (3o) relationships as follows:

KNOWN FACTORS: R, e, P, θO_(p) UNKNOWN FACTORS: r₁, α_(p), ( α_(p1), α₂,α₃, α₄ )

    1. α.sub.p = θ.sub.p - α.sub.2

    2. α.sub.p1 = (P + 1 ) . α.sub.p

    3. α.sub.2 = θ.sub.p - α.sub.p

    4. α.sub.3 = P . α.sub.p

    5. α.sub.4 = α.sub.p1 - θ.sub.p

    6. x.sub.1 = R . cos α.sub.p

    7. x.sub.2 = e . cos α.sub.p1

    8. y.sub.1 = R . sin α.sub.p

    9. y.sub.2 = e . sin α.sub.p1

    10. x.sub.p = x.sub.1 + x.sub.2

    11. y.sub.p = y.sub.1 + y.sub.2

    12. x.sub.p = R . cos α.sub.p + e . cos α.sub.p1

    13. y.sub.p = R . sin α.sub.p + e . sin α.sub.p1

    14. r.sub.p = √(x.sub.1 + x.sub.2 ).sup.2 + (y.sub.1 + y.sub.2 ).sup.2

    15. r.sub.p = √x.sub.p.sup.2 + y.sub.p.sup.2

    16. r.sub.p = (R+e)-[R.(1-cos α.sub.2)+e.(1-cos α.sub.4)]

    17. r.sub.p = R . cos α.sub.2 + e . cos α.sub.4 ##EQU27##

    23. r.sub.p.sup.2 = (R + cos α.sub.3 . e ).sup.2 + (sin α.sub.3 . e ).sup.2

    24. r.sub.p.sup.2 = R.sup.2 + 2 . e . (x.sub.p.sup.2 + y.sub.p.sup.2).sup.1/2 cos α.sub.4 - e.sup.2

    25. r.sub.p.sup.2 = e.sup.2 + 2 . R (x.sub.p.sup.2 + y.sub.p.sup.2).sup.1/2 cos α.sub.2 - R.sup.2 ##EQU28##

These relationships are developed for designing purposes and areaccurate, as will be apparent to the reader from further consideration.

The following relationships disclose the area A_(p) of the rotarypiston: ##EQU29##

The following relationships describe the area A_(ps) of the circularsection of the rotary piston: ##EQU30## Where

    θ.sub.p = Polar angle.

    θ.sub.1p = Circular sectors angle, lower limit.

    θ.sub.2p = Circular sectors angle, upper limit.

The following describes the lengths of curve profile S_(p) of the rotarypiston: ##EQU31##

The following equation describes the length of curve profile S_(ps) ofthe circular section: ##EQU32##

The relationships below discloses the total area A_(w) of all of theworking chambers created by the rotation in the stationary housing ofthe rotary piston engine with multi-explosion chambers:

    A.sub.w = A.sub.h - A.sub.p                                (104 )

Where

    A.sub.w = Total area of all of the working chambers.

    A.sub.h = Area of the stationary housing.

    A.sub.p = Area of the rotary piston.

The following equations disclose minimum area A_(min) of any of theworking chambers, created by the rotary piston in determined positionsin the stationary housing. In order to develop the minimum area A_(min),the coordinate axes x_(h) and y_(h) of the stationary housing curveprofile must be translated. To understand the mode of the translationsof the coordinate axes x_(h) and y_(h), reference is made to FIG. 4wherein the coordinate axes x_(p) and y_(p) of the rotary piston curveprofile and the coordinate axes x_(h) and y_(h), x_(h2) and y_(h2),x_(h3) and y_(h3) of the statonary housing curve profile areillustrated.

The following equations disclose the relationship between the coordinateequations of the rotary piston curve profile and the stationary housingcurve profile. These are the basic coordinate equations:

    x.sub.p = cos α.sub.p . R + cos (P + 1 ) α.sub.p . e (105 )

    y.sub.p = sin α.sub.p . R + sin (P + 1) α.sub.p . e (106 )

    x.sub.h = cos α.sub.h . (R + e ) + cos (H + 1 ) α.sub.h . e (107 )

    y.sub.h = sin α.sub.h . (R + e ) + sin (H + 1 ) α.sub.h . e (108 )

As part of the previously mentioned translation, the coordinate axesx_(h) and y_(h) of the stationary housing curve profile will betranslated to coordinate axes x_(h2) and y_(h2) about an angle γ. Theangle γ is a translational angle defined as the twist of the x_(h2),y_(h2) frame of reference with respect to the x_(h) and y_(h) frame ofreference. This will result in a twsit translation of the coordinatedaxes x_(h) and y_(h) to the coordinate axes x_(h2) and y_(h2), andtransformation of the coordinate equations x_(h) and y_(h) to thecoordinate equations x_(h2) and y_(h2), as follows: ##EQU33##

    x.sub.h2 = cos α.sub.h . (R + e ) - cos (H + 1 ) α.sub.h . e (110 )

    y.sub.h2 = sin α.sub.h . (R + e ) - sin (H + 1 ) α.sub.h . e (111 )

Where:

γ = Translation angle between the coordinate axes x_(h) and x_(h2)respectively y_(h) and y_(h2) of the stationary housing curve profile,

H = Number of the geometrical convex recesses of the stationary housing;

x_(h2) = Transformed coordinate equation of x_(h) by the twisttranslation of the coordinate axis x_(h) to the coordinate axis x_(h2) ;

y_(h2) = Transformed coordinate equation of y_(h) by the twisttranslation of the coordinate axis to the coordinate axis y_(h2).

As another part of the previously mentioned translations, the origin Oof the stationary housing curve profile will be translated along anangle γ₁ and with a distance e to the origin 0₁ of the rotary pistoncurve profile. This will result in a linear translation of thecoordinate axis x_(h2) to the coordinate axis x_(h3) and a paralleltranslation of the coordinate axis y_(h2) to the coordinate axis y_(h3)and transformation of the coordinate equations x_(h2) and y_(h2) to thecoordinate equations x_(H3) and y_(h3), as follows:

    γ.sub.1 = 180° = constant                     (112)

    x.sub.h3 = x.sub.h2 + cos γ.sub.1 . e                (113)

    y.sub.h3 = y.sub.h2 . cos γ.sub.1                    (114)

Where:

γ₁ = Translation angle between the origin 0 of the stationary housingcurve profile and the origin 0₁ of the rotary piston curve profile;

e = Eccentricity of the rotary piston;

x_(h3) = Transformed coordinate equation of x_(h2), by the lineartranslation of the coordinate axis x_(h2) to the coordinate axis x_(h3);

y_(h3) = Transformed coordinate equation of y_(h2), by the paralleltranslation of the coordinate axis y_(h2) to the coordinate axis y_(h3).

Further relationships may be developed as follows. These equationsassist to determine the lower limit of integration. ##EQU34##

    γ.sub.4 = γ.sub.2 - γ.sub.3              (117)

Where:

γ₂ = Half of the actual angle between the center line of the rotarypiston lobes;

γ₃ = A suitable angle between the outside of the S seal strip (23) andthe center line of the rotary piston lobe;

γ₄ = Half of the effective circular sections angle;

P = Number of the geometrical convex lobes of the rotary piston;

S_(D) = Seal strips S(23) outside distance from the center line of therotary piston lobes;

R = Rotary piston curve profile generating radius;

e = Eccentricity of the rotary piston.

The following equation describes the lower limits β of integration ofthe function x_(h3) ² + y_(h3) ² of the stationary housing circularsection area and the function x_(p) ² + y_(p) ² of the rotary pistoncircular section area.

    β = γ.sub.1 - γ.sub.4                     (118)

Where:

β = Lower limits of integrations of the function x_(h3) ² + y_(h3) ² ofthe stationary housing circular section area and the function x_(p) ² +y_(p) ² of the rotary piston circular section area.

The following explanation and corresponding relationships disclose howto describe in the minimum area A_(min) of any of the working chamberscreated by the eccentrically rotated rotary piston curve profilesurface, limited by the seal strips S(23). From the precedingexplanation and the relevant equations, it is evident that a pointp_(h3) (x_(h3), y_(h3)) of the stationary housing curve profile is afunction of x_(h3) and y_(h3). The foregoing equations disclose thepositions of the point p_(h3) (x_(h3), y_(h3)) which in fact forms atriangle with abscissa x_(h3), ordinate y_(h3) or can be described by ahypotenuse or polar radius r_(h3), with an angle θ_(h3) relative to theorigin 0₁. The relationships of this triangle are the following:##EQU35##

    r.sub.h3 = √ x.sub.h3.sup.2 + y.sub.h3.sup.2        (123)

Where:

θ_(h3) = Polar angle;

r_(h3) = hypotenuse or polar radius;

x_(h3) = Abscissa, function;

y_(h3) = Ordinate, function. ##EQU36##

The following equations describe the maximum area A_(max) of any of theworking chambers created by the rotary piston in determined positionsand by the eccentric rotations of the rotary piston in the stationaryhousing. In order to describe the maximum area A_(max), the coordinateaxes x_(h) and y_(h) of the stationary housing curve profile must betranslated. To better understand the mode of this translation ofcoordinate axes x_(h) and y_(h) , reference is made to FIG. 1 where thecoordinate axes x_(p) and y_(p) of the rotary profile and the coordinateaxes x_(h) and y_(h), x_(h1) and y_(h1) of the stationary housing curveprofile are illustrated.

As part of the previously mentioned translation of the coordinate axesx_(h) and y_(h), the origin O of the stationary housing curve profilewill be translated along an angle γ₁ and with a distance e to the originO₁ of the rotary piston curve profile. This will result a lineartranslation of the coordinate axis x_(h) to the coordinate axis x_(h1)and in a parallel translation of the coordinate axis y_(h) to thecoordinate axis y_(h1), and transformation of the coordinate equationsx_(h) and y_(h) to the coordinate equations x_(h1) and y_(h1) asfollows:

    x.sub.h1 = x.sub.h - cos γ.sub.1 . e                 (128)

    y.sub.h1 = y.sub.h . cos γ.sub.1                     (129)

Where:

γ₁ = Translation angle between the origin O of the stationary housingcurve profile and the origin O₁ of the rotary piston curve profile;

x_(h1) = Transformed coordinate equation of x_(h), by the lineartranslation of the coordinate axis x_(h) to the coordinate axis x_(h1).

y_(h1) = Transformed coordinate equation of y_(h), by the paralleltranslation of the coordinate axis y_(h) to the coordinate axis y_(h1).

The following explanation and corresponding relationships describe themaximum area A_(max) of any of the working chambers created by theeccentrically rotated rotary piston curve profile surface and limited bythe seal strips S(23) within the stationary housing curve profilesurface. From the preceding explanation and corresponding relationshipsit is evident that a point P_(h1) (x_(h1), y_(h1)) on the stationaryhousing curve profile is a function of x_(h1) and y_(h1). The foregoingrelationships describe the positions of the point P_(h1) (x_(h1),y_(h1)) which in fact forms a triangle with abscissa x_(h1), ordinatey_(h1), and may be described by an hypotenuse of polar radius r_(h1),with an angle θ_(h1) relative to the origin O₁. The relationships ofthis triangle are the following: ##EQU37##

    r.sub.h1 √ x.sub.h1.sup.2 + y.sub.h1.sup.2          (134)

Where:

θ_(h1) = Polar angle

r_(h1) = Hypotenuse or polar radius;

x_(h1) = Abscissa, function;

y_(h1) = Ordinate, function. ##EQU38##

From the preceeding coordinate relationships it is possible to determinethe area A_(h), the circular section area A_(hs), the length of curveprofile S_(h) and the circular section of curve profile lenth S_(hs) ofthe stationary housing of a rotary piston engine with multi explosionchambers. Further, the preceeding relationships demonstrate also how todetermine the area A_(p), the circular section area A_(ps), the lengthof the curve profile S_(p) and the circular section curve profile lengthS_(ps) of the rotary piston.

Moreover, the preceeding relationships also describe how to determinethe minimum area A_(min), and the maximum area A_(max) of the workingchambers of a rotary piston engine with multi explosion chambers. Thedevelopment of these coordinate equations was based on the fact that thenumber of the geometrical convex recesses H was an even number. As aresult of the development of these coordinate equations, the number ofthe geometrical convex lobes P was an odd number.

If the number of geometrical convex recesses H₁ is an odd number, thenthe basic coordinate equations will be as follows:

    x.sub.hu = cos α.sub.hu . R.sub.1 - cos α.sub.ha1 . e (139)

    x.sub.hu = cos α.sub.hu . (R + e) - cos (H.sub.1 + 1) α.sub.hu . e                                                       (140)

    y.sub.hu = sin α.sub.hu . R.sub.1 - sin α.sub.hu1 . e (141)

    y.sub.hu = sin α.sub.hu . (R + e) - sin (H.sub.1 + 1) α.sub.hu . e                                                       (142)

As a result of the development of the above coordinate equations, thenumber of the geometrical convex lobes P₁ will be an even number and thebasic coordinate equations will be as follows:

    x.sub.pu = cos α.sub.pu . R - cos α.sub.pu1 . e (143)

    x.sub.pu = cos α.sub.pu . R - cos (P.sub.1 + 1) α.sub.pu . e (144)

    y.sub.pu = sin α.sub.pu . R - sin α.sub.pu1 . e (145)

    y.sub.pu = sin α.sub.pu . R - sin (P.sub.1 + 1) α.sub.pu . e (146)

The following relationships describe how to determine the minimumV_(min) of any of the working chambers created by rotation of the pistonwithin the stationary housing.

    V.sub.min = A.sub.min . W.sub.h                            (147) ##EQU39## Where: V.sub.min = Minimum volume of any of the working chambers;

A_(min) = Minimum area of any of the working chambers;

W_(h) = Stationary housing width;

R = Rotary piston curve profile generating radius;

e = Eccentricity of the rotary piston.

The following relationships describe the maximum volume V_(max) of anyof the working chambers created by rotation of the piston within thestationary housing.

    V.sub.max = A.sub.max . W.sub.p                            (149) ##EQU40## Where: V.sub.max = Maximum volume of any of the working chambers;

A_(max) = Maximum area of any of the working chambers;

W_(p) = Rotary piston width;

R = Rotary piston curve profile generating radius;

e = Eccentricity of the rotary piston.

The gearing system of this invention is described in detail in my U.S.Pat. No. 3,771,501, previously discussed. The details of U.S. Pat. No.3,771,501 are hereby incorporated by reference into the disclosure ofthis invention in toto. Column 2, lines 14-47 clearly describe thegearing system that will be dealt with in exact detail below.

For reference, it will be appreciated from a review of U.S. Pat. No.3,771,501 that:

D_(h) = Diameter of internal gear element 29;

D_(p) = Diameter of internal gear 28 affixed to the rotary piston;

D_(c) = Diameter of planetary gear element 25;

D_(s) = Diameter of output shaft gear element 24.

The transverse cross-sectional assembly view of the gearing system (FIG.4) discloses the gearing layout of the rotary piston engine with multiexplosion chambers. The gearing system has been developed to transmitpower from the eccentrically rotated rotary piston to the output shaft.The gearing relationships will now be described and explained below:

    D.sub.h = 2 . H . e                                        (151)

    D.sub.h = D.sub.p +2 . e                                   (152)

    D.sub.p = 2 . 2 . P . e                                    (153)

    D.sub.p = D.sub.h - 2 . e                                  (154)

    D.sub.c = D.sub.h /3                                       (155) ##EQU41##

    D.sub.s = D.sub.h /3                                       (157) ##EQU42##

    D.sub.h /D.sub.p = H/P = R.sub.G                           (159)

    D.sub.h /D.sub.c = 3/1 = R.sub.T = R.sub.O                 (160)

    d.sub.h /D.sub.s = 3/1 = R.sub.O = R.sub.T                 (161) ##EQU43##

    D.sub.e = 2 . e                                            (168)

Where:

D_(h) = Stationary housings internal gear diameter;

D_(p) = Rotary pistons internal gear diameter;

D_(c) = Cam tracks toothed sprocket diameter;

D_(s) = Output shafts fixed toothed sprocket diameter;

H = Number of the geometrical convex corners of the stationary housing;

P = Number of the geometrical convex corners of the rotary piston;

R_(G) = The ratio D_(h) /D.sub. p

R_(T) = The ratio D_(h) /D.sub. s

R_(o) = the ratio D_(h) /D_(c) ;

R_(p) = e = Radius from the output shaft axis to the center of gravityof the rotary piston for various positions of the rotary piston;

e = R_(p) = Eccentricity of the rotary piston.

D_(e) = Diameter of eccentricity.

From the foregoing it is clear that the gear system can be reduced tothe following three relationships for the purpose of determining thevalues of Dh, Dp, Ds and DC:

    i. d.sub.h =  2 . H. e

    II. D.sub.h =  2 . p . e ##EQU44##

The above represents a system of three equations in seven unknowns andin which two of the unknowns are equal, thereby making the effectivenumber of unknowns equal to six. It is thereby obvious that if any threeof the unknowns are known or given, then the other three can be solvedfor by means of the foregoing simultaneous equations.

In discussing the practical limits of this invention it is useful toappreciate the value of the limitation factor L. L is defined as:

    L = √E/K ≧ 1/1 < 1.5 / 1                     (169)

where:

L = Limitation factor;

E = Compression ratio (previously discussed);

K = Ratio factor (previously discussed);

In other words:

    1 ≦ L ≦ 1.5                                  (170)

it will be appreciated from the foregoing that the compression ratio √Eis proportional to L and that:

    1 ≦ √E/K ≦ 1.5                        (171)

it follows then that the compression ratio is also proportional to thechoice of R and e.

Of importance also are the following relationships:

    C.sub.h =  K/H ≧ 1/1 < 2.75/1                       (172)

    c.sub.p = k.sub.2 /p ≧ 1/1 < 3/1                    (173)

or

    2.75 > C.sub.h ≧ 1                                  (174)

    3 > C.sub.p ≧ 1                                     (175)

Where:

C_(h) = Curve factor of the stationary housing curve profile C_(o) ;

C_(p) = Curve factor of the rotary piston curve profile C₁.

The foregoing relationships disclose the geometrical structure of therotary piston engine with multiexplosion chambers according to theinvention.

According to this invention, the rotary piston engine with multiexplosion chambers can be developed with the required H number of lobesor convex recesses. If the number H is an even number, then the rotarypiston engine will preferably have H/2 explosion chambers. In the eventthat the number H is an odd number, then the rotary piston engine willpreferably have H explosion chambers. In all cases, the number H may belimited because of the technical and economical considerations.

It is to be further understood that additional forms of this inventionrequiring the use of more dual explosion chambers and variations in theshape of the stationary housing and the rotary piston are within thescope of this invention.

In a general manner, while there has been disclosed effective andefficient embodiments of the invention it should be well understood thatthe invention is not limited to such embodiments as there might bechanges made in the arrangement, disposition and form of the partswithout departing from the principle of the present invention ascomprehended within the scope of the accompanying claims.

I claim:
 1. An improved multilobe rotary piston engine comprising:ahousing including H lobes and having an interior cavity profiledescribed by the curve generated by the locus of a point P_(h) withinthe diameter D₁ of a rolling circle as it revolves around a second fixedcircle of diameter D₀ ; and a piston having P lobes and a profile whichpermits operative engagement of the piston with the profile of thehousing profile, said piston having a profile described by a curvegenerated by the locus of a point P_(p) within the diameter D₃ of arolling circle as it revolves around another fixed circle of a diameterD₂ ; wherein the relationship between the profile of the housing and theprofile of the piston is described as follows:

    a. H = P + 1

    b. H = D.sub.0 /D.sub. 1

    c. P = D.sub.2 /D.sub.3 ##EQU45## Where: H = The number of lobes on the stationary housing;

P = The number of lobes on the piston; R = Minimum interior radius ofthe housing; e = The eccentricity of the rotary piston; D₀ = Fixedcircle diameter of the stationary housing; D₁ = Rolling circle diameterof the stationary housing; D₂ = Fixed circle diameter for the rotarypiston; and D₃ =Rolling circle diameter for the rotary piston; whereinsaid engine is further limited to operation in the following range of L:

    ≦  l ≦ 1.5; and,

    L = √E/K

Where: L = Limitation range factor; E = the engine compression ratio;and, K = R/e .
 2. An improved multilobe rotary piston enginecomprising:a housing including H lobes and having an interior cavityprofile described by the curve generated by the locus of a point P_(h)within the diameter D₁ of a rolling circle as it revolves around asecond fixed circle of diameter D₀ ; and a piston having P lobes and aprofile which permits operative engagement of the piston with theprofile which permits operative engagement of the piston with theprofile of the housing profile, said piston having a profile describedby a curve generated by the locus of a point P_(p) within the diameterD₃ of a rolling circle as it revolves around another fixed circle ofdiameter D₂ ; wherein the relationship between the profile of thehousing and the profile of the piston is described as follows:

    a. H = P + 1

    b. H = D.sub.0 /D.sub.1

    c. P = D.sub.2 /D.sub.3 ##EQU46## Where: H = The number of lobes on the stationary housing;

P = The number of lobes on the piston; R = Minimum interior radius ofthe housing; e = The eccentricity of the rotary piston; D₀ = Fixedcircle diameter of the stationary housing; D₁ = Rolling circle diameterof the stationary housing; D₂ = Fixed circle diameter for the rotarypiston; and D₃ = Rolling circle diameter for the rotary piston;whereinsaid engine is further limited to operation in the following range ofC_(h) and C_(p) :
 2. 75 > C_(h) ≧ 1; and,

    3 > C.sub.p ≧ 1

Where: ##EQU47##

    K = R/e; and, ##EQU48##


3. An improved multilobe rotary piston engine comprising: a housingincluding H lobes and having an interior cavity profile described by thecurve generated by the locus of a point P_(h) within the diameter D₁ ofa rolling circle as it revolves around a second fixed circle of diameterD₀ ; anda piston having P lobes and a profile which permits operativeengagement of the piston with the profile of the housing profile, saidpiston having a profile described by a curve generated by the locus of apoint P.sub. within the diameter D₃ of a rolling circle as it revolvesaround another fixes circle of diameter D₂ ; wherein the relationshipbetween the profile of the housing and the profile of the piston isdescribed as follows:

    a. H = P + 1

    b. H = D.sub.0 /D.sub.1

    c. P = D.sub.2 /D.sub. 3 ##EQU49## Where: H = The number of lobes on the stationary housing;

P = The number of lobes on the piston; R = Minimum interior radius ofthe housing; e = The eccentricity of the rotary piston; D₀ = Fixedcircle diameter of the stationary housing; D₁ = Rolling circle diameterof the stationary housing; D₂ = Fixed circle diameter for the rotarypiston; and D₃ = Rolling circle diameter for the rotary piston; saidengine further including a planetary gearing system comprising: aninternal gear G_(h) of diameter D_(h) affixed firmly to the stationaryhousing; an internal gear G_(p) of diameter D_(p) affixes firmly to thepiston; a planetary gear D_(c) of diameter D_(c) operatively engagingboth the internal gears G_(h) and G_(p) ; and an output shaft driveD_(s) of diameter D_(s) operatively engaged with said planatary gear;wherein the diameters of D_(h), D_(p), D_(c) and D_(s) are related asfollows:

    a. D.sub.h = 2 . H . e

    b. D.sub.p = 2 . P . e and ##EQU50## Where: D.sub.h = Stationary housing internal gear diameter;

D_(p) = Rotary piston internal gear diameter; D_(c) = Planetary gear camtoothed sprocket diameter; and D_(s) = Output shaft fixed tooth sprocketdiameter.
 4. The invention of claim 3 wherein the curve profile of thehousing is an epitrochoid.
 5. The invention of claim 4 wherein the curveprofile of the piston is an epitrochoid.
 6. The invention of claim 4wherein H ≦ 7.